## Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics

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Ginzburg, Viktor L.

Gurel, Basak Z.

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The main theme of this paper is the connection between the existence of
infinitely many periodic orbits for a Hamiltonian system and the behavior of
its action or index spectrum under iterations. We use the action and index
spectra to show that any Hamiltonian diffeomorphism of a closed, rational
manifold with zero first Chern class has infinitely many periodic orbits and
that, for a general rational manifold, the number of geometrically distinct
periodic orbits is bounded from below by the ratio of the minimal Chern number
and half of the dimension. These generalizations of the Conley conjecture
follow from another result proved here asserting that a Hamiltonian
diffeomorphism with a symplectically degenerate maximum on a closed rational
manifold has infinitely many periodic orbits.
We also show that for a broad class of manifolds and/or Hamiltonian
diffeomorphisms the minimal action--index gap remains bounded for some infinite
sequence of iterations and, as a consequence, whenever a Hamiltonian
diffeomorphism has finitely many periodic orbits, the actions and mean indices
of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian
diffeomorphisms of the n-dimensional complex projective space with exactly n+1
periodic orbits a stronger result holds. Namely, for such a Hamiltonian
diffeomorphism, the difference of the action and the mean index on a periodic
orbit is independent of the orbit, provided that the symplectic structure on
the projective space is normalized to be in the same cohomology class as the
first Chern class.

Comment: 46 pages, 1 figure

Comment: 46 pages, 1 figure

##### Keywords

Mathematics - Symplectic Geometry, Mathematics - Dynamical Systems, 53D40, 37J45, 70H12